On the theory of extraction from falling droplets II

Abstract

SumrnaryThe theory of extraction of a substance, dissolved in droplets of a fluid. when these are falling (or rising) in an other fluid under the influence of gravity. is specialized for the case that the rate of falling (or rising) is very small. A perturbation method, well-known from quantum mechanics, is applied to determine the time of diffusion in second approximation. § 1. Introduction. In a paper by K ron i g and B r ink 1), to be quoted in this article as part I, the extraction by the surrounding medium of a substance, dissolved in a droplet falling or rising under the influence of gravity, was investigated theoretically. The calculations presented there were specialized for the case that, the diffusion coefficient D 2 of the dissolved substance in the surrounding medium is large compared with the diffusion coefficient D 1 in the droplet so that the concentration at the surface.of the droplet may be taken as vanishingly small. Moreover the velocity of the droplet was assumed to be so large that the average circulation time t, in the vortices set up in the droplet by its motion is small compared with the diffusion time t a of the stationary droplet, the latter being defined as the time in which the total mass of dissolved substance in the droplet, originally spread uniformly over its volume, falls to lie of its initial value.In the present paper we wish to discuss the opposite case, viz. that the droplet falls or rises so slowly as to make i,~t a . In that case the concentration, considered as a function of the coordinates and the time, will differ only slightly from the concentration in a stationary droplet and this suggests the application of perturbation methods. Such perturbation methods, in, fact, have been developed in con--103-